How to select edge rings between two edges?
It may be easier to use another selection tool instead of Edge Ring selection, because it won't take into account specified end of selection.

You can workaround that by selecting faces:


Select the shortest path with Path Selection in Face Select mode.   
Switch to Edge Select mode, and deselect top and bottom edge loops by holding Alt+Shift and clicking with LMB on them. "Shortened" edge ring will be the only left selected.